i. Ax=b is inconsistent (i.e., no solution exists) if and only if rank[A]<rank[A∣b].
ii. Ax=b has an unique solution if and only if rank[A]=rank[A∣b]=n.
iii. Ax=b has infinitely many solutions if and only if rank[A]=rank[A∣b]<n.
A=⎣⎡012−64−501−34⎦⎤ Swap rows 1and 2
⎣⎡120−6−540−314⎦⎤R3=R3+(1/2)R1
⎣⎡1200−54−5/2−315/2⎦⎤ R3=R3+(5/8)R2
The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is 3.
Since rank[A]=rank[A∣b]=3=n, then Ax=b has an unique solution.
Therefore, the given system of equations is consistent.
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